Mechanical systems exhibit special modes of vibration in which all parts of the system move at the same frequency. These special vibrations are called normal modes. Each normal mode has its own characteristic frequency. All possible motions of the system consist of mixtures of these modes thus all sounds produced by the system will have spectra made up from the characteristic frequencies. In order for an instrument to produce a well tuned sound the characteristic frequencies of its tone produing part must form an harmonic series.
Simple systems of masses and springs have as many normal modes as they have masses. More complex systems, strings, bars, plates, tubes, etc., can exhibit very large numbers of normal modes.
The normal modes of a thin string clamped at its ends form an harmonic series whose fundamental frequency depends on the speed of sound on the string and on the length of the string. The frequencies are all of the form
where v is the speed of sound on the string and L is the length of the string. n is the mode number. Mode number n has n anti-nodes spaced evenly along the string.
When a string is plucked its spectrum depends on the point at which it is plucked.
Plucking near the middle principally excites the first and third modes of the string
and produces a rather hollow sound. Plucking nearer to one end of the string
increases the amount of the higher frequency modes and produces a brighter sound.
Once a string has been plucked the sound energy is communicated by the bridge to
the body of the instrument which vibrates in sympathy, amplifying the sound and
drawing energy out of the vibrating string. The more efficiently the body vibrates,
the louder the sound and the faster the sound dies away. Thus banjos play loud
notes for a very short time, acoustic guitars play quieter notes that last quite a lot
longer, and electric guitars, whose bodies hardly vibrate at all, produce almost no
sound but the strings vibrate for a very long time (minutes).
The note that a string produces can be altered by altering the effective length of the string, altering the mode in which the string vibrates, or altering the tension in the string.
Westen stringed instruments normallyoperate on their fundamental mode and alter the note by changing the sounding length of the string by pressing the string down to a fingerboard with or without frets. Different notes can also be produced by lightly touching the string at node points to force the string to vibrate in a different mode. Some non-western bowed strings are played exclusively by altering the mode. A few stringed instruments also allow bending pitches by altering the tension in the string to provide an additional musical effect. This is much more effective with metal strings than with nylon strings.